# Describe the region of R^3, sphere with inequality

1. Sep 23, 2011

### alias

1. The problem statement, all variables and given/known data

Describe the region of R^3 that is represented by:

2. Relevant equations

x^2 + y^2 + z^2 > 2z

3. The attempt at a solution

I'm not sure what to do with this at, especially at z=0 and z=2

2. Sep 23, 2011

### flyingpig

Maybe you should complete the square

3. Sep 23, 2011

### alias

x^2 + y^2 + (z-1)^2 > 1
a sphere with centre (0,0,1) and radius >1?

4. Sep 23, 2011

### SammyS

Staff Emeritus
Seems like that could be many spheres.

5. Sep 23, 2011

### alias

would it be:
x^2 + y^2 + (z-1)^2 > 1

infinite number of spheres with centre (0,0,1) and radius >1?

6. Sep 23, 2011

### SammyS

Staff Emeritus
Where do you find any point (x, y) that satisfies x^2 + y^2 + (z-1)^2 > 1 , relative to the set of points that satisfy x^2 + y^2 + (z-1)^2 = 1 ?

7. Sep 24, 2011

### alias

I'm lost now. I am fairly sure I have the centre correct at (0,0,1), the only thing I can think of regarding your previous post is:
(x,y) = (x>1, y>1) or
(x,y) = (x>=1, y>=1). But this seems incorrect.

8. Sep 24, 2011

### SammyS

Staff Emeritus
Yes, the equation x2 + y2 + (z-1)2 = 1 describes a sphere of radius 1 with center at (x, y, z) = (0, 0, 1) .

Pick any point that is outside of the sphere. What do you get for x2 + y2 + (z-1)2 for that point?

Pick any point that is inside of the sphere. What do you get for x2 + y2 + (z-1)2 for this point?

9. Sep 25, 2011

### alias

for the equation, a point inside the sphere would be (1/2, 0, 1)
.5^2 + 0^2 + (1-1)^2 = .25

outside the the sphere would be (1,1,1)
1^2 + 1^2 + (1-1)^2 = 2

10. Sep 25, 2011

### SammyS

Staff Emeritus
Notice that .25 < 1, so for point (1/2, 0, 1), x2 + y2 +(z-1)2 < 1 .

Similarly, 2 > 1, , so for point (1, 1, 1), x2 + y2 +(z-1)2 > 1 .

Does that give you any ideas?

BTW, what does $\sqrt{(x-0)^2 + (y-0)^2 + (z-1)^2}$ represent ?

11. Sep 25, 2011

### alias

\sqrt{(x-0)^2 + (y-0)^2 + (z-1)^2}
is the distance between two points, (0, 0, -1) and any point (x,y,z)
but I still don't understand the constraint on x, y, and/or z.

12. Sep 25, 2011

### alias

or the inequality of the radius

13. Sep 25, 2011

### HallsofIvy

I'm afraid SammyS rather confused things when he asked about "(x, y)" in post #6.

alias, In post #9, you said, "outside the the sphere would be (1,1,1)". Yes, that is a point in this set but any point on that set lies in this set. The only problem with your post #5, "infinite number of spheres with centre (0,0,1) and radius >1?", is not enough. The set of all spheres with integer radius, greater than 1, satisfies that but does not include points that satisfy, for example, $x^2+ y^2= \frac{9}{4}$.

14. Sep 25, 2011

### SammyS

Staff Emeritus
Yes. TYPO !!!

I meant to type " point (x, y, z) = ... "

15. Sep 25, 2011

### SammyS

Staff Emeritus
Actually, it's the distance between two points, (0, 0, 1) and any point (x,y,z). In other words, it's the distance from the center of your sphere to any point (x,y,z).

If $\sqrt{(x-0)^2 + (y-0)^2 + (z-1)^2}>1\,,$ where is the point (x, y, z) located in relation to the sphere ?

16. Sep 26, 2011

### alias

The distance between (0,0,1) and (x,y,z) is greater than 1. I'm not sure about the relation to the sphere though.

17. Sep 26, 2011

### SammyS

Staff Emeritus
What is the distance from (0,0,1) to any point on the surface of the sphere ?

18. Sep 26, 2011

### alias

The distance from (0,0,1) to any point on the surface of the sphere is 1.
How does this relate to the inequality with the radius?

19. Sep 26, 2011

### SammyS

Staff Emeritus
If the distance from (0,0,1) is 1, the point (x,y,z) is on the surface of the sphere. If that distance is greater than 1 doesn't the point lies outside the sphere?

If $(x)^2 + (y)^2 + (z-1)^2>1\,,$ what does that say about $\sqrt{(x-0)^2 + (y-0)^2 + (z-1)^2}\,?$

20. Sep 26, 2011

### alias

Yes, any point (x,y,z) that has a distance greater than 1 from (0,0,1) lies outside the the sphere. So the original eqn, x^2 + y^2 + (z-1)^2 > 1 consists of all points outside the outside the sphere with with centre (0,0,1) and radius 1? Would that not be all spheres with radius >1 and the same centre?